## 1. Introduction

In Cantarello et al. (2022, hereafter Part I), we presented and discussed both the dynamics and the numerics of a new idealized model (“ismodRSW”) to be used in future satellite data assimilation (DA) experiments. In this paper, or Part II, we show a formal mathematical derivation of the underlying isentropic 1½-layer shallow-water model based on variational principles and Hamiltonian fluid dynamics.

Shallow-water models represent a class of simplified fluid-dynamic models often utilized to describe analytically and numerically a number of fundamental and theoretical properties of stratified fluids, including the effect of rotation (e.g., as in the Rossby adjustment problem) and the propagation of gravity waves. In this regard, Zeitlin (2018) provides a broad overview of the use of shallow-water models in geophysics, including “moist” isentropic models able to mimic convection and precipitation. In recent decades, shallow-water models have also been utilized as idealized tools in DA research, for both oceanic and atmospheric applications (Žagar et al. 2004; Salman et al. 2006; Stewart et al. 2013; Würsch and Craig 2014; Kent et al. 2017).

*g*and a

*reduced*gravity

*g*′ is introduced:

*ρ*

_{1}and

*ρ*

_{2}indicate the densities of the fluid in the upper and lower layer, with the least dense layer on top, i.e.,

*ρ*

_{2}>

*ρ*

_{1}. Instead, moving from an isopycnal model (constant density) to an isentropic one (constant potential temperature) leads to a different set of equations which, more importantly, are valid in a different atmospheric regime.

A “dry” isentropic 1½-layer shallow-water model (without convection and precipitation) should naturally arise from an isentropic two-layer model after imposing a rigid-lid condition on the top layer. Here, starting from the isentropic *N* = 2-layer model derived by Bokhove and Oliver (2009), we show that this approach leads to an apparent inconsistency in the model equations, in which a zero Montgomery potential constraint (*M*_{1} = 0) seems not to be preserved in time by the continuity equations of the layers. To resolve this contradiction, we adopt principles of Hamiltonian fluid dynamics (exploiting a slaved Hamiltonian approach) and introduce fast and slow variables (Van Kampen 1985) arising from an asymptotic analysis performed on an isentropic two-layer shallow-water model. Crucially, we will show that this asymptotic analysis relies on a series of scaling assumptions that can be justified on the basis of real-word observations obtained from radiosonde data, in the presence of low-level jet (LLJ) conditions. In the end, we show that the rigid-lid condition (*M*_{1} = 0) needs to be accompanied by fluid at rest in the top layer (**v**_{1} = 0) for the isentropic 1½-layer model to be a consistent approximation of a two-layer one.

The derivation of balanced fluid dynamical models exploiting Hamilton’s principle in which high-frequency waves are filtered out started with the work of Salmon (1983, 1985, 1988). In particular, the use of Dirac brackets’ theory (Dirac 1958, 1964) applied to the Hamiltonian derivation of multilayer shallow-water models was developed further in Bokhove (2002a) and Vanneste and Bokhove (2002). The derivation of an *N*-layer isentropic shallow-water model based on Hamiltonian mechanics was given in Bokhove and Oliver (2009) and will constitute the starting point of our study. In this regard, the reader might find useful to know that parts of the work treated in this paper has appeared in a previously unpublished manuscript (Bokhove 2007).

The structure of the paper is as follows. In section 2, we will start with presenting the equations of a full two-layer isentropic model and show how imposing a rigid-lid condition leads to a seemingly inconsistent yet closed 1½-layer model. In section 3 we introduce a scaling for the two-layer model and its equations are subsequently nondimensionalized; afterward, an asymptotic analysis based on the method of multiple time scales is conducted. In section 4 we use radiosonde observations to justify the scaling used in the asymptotic analysis. In section 5, the Hamiltonian derivation of the isentropic 1½-layer shallow-water model is discussed. Conclusions are given in section 6.

## 2. A rigid-lid approximation in a two-layer model

We start this section by presenting an isentropic two-layer model and finish with an argument how a closed 1½-layer model emerges by taking a seemingly inconsistent rigid-lid approximation. That the final model is nonetheless consistent will be subsequently shown in a combined asymptotic and Hamiltonian analysis, resulting in a rigid-lid condition with a (nearly) passive and high upper layer. The Hamiltonian derivation demonstrates that the 1½-layer model has a bona fide conservative and hyperbolic structure, which is exploited in the numerical discretization discussed in Part I.

*N*–layer model can be found in Bokhove and Oliver (2009). Here, we take a two-layer simplification thereof, with

*N*= 2. Figures 1a and 1c provide a sketch of the two-layer model configuration. The momentum equations of the model arise by assuming hydrostatic balance and constant entropy (potential temperature

*θ*) in each layer. The continuity equations emerge once the space (

*x*,

*y*) and time-dependent (

*t*) pseudodensity

*σ*(

_{α}*x*,

*y*,

*t*) for each layer, numbered by

*α*= 1, 2, is defined, i.e.,

*g*refers to the gravity acceleration and

*η*−

_{α}*η*

_{α}_{−1}is the net nondimensional pressure difference between the bottom and the top of the layer

*α*, with

*η*defined as

*η*=

*p*/

*p*for a reference pressure

_{r}*p*. The pseudodensity

_{r}*σ*arises from hydrostatic balance

*dp*= −

*ρgdz*, integrating an element of mass flux for some infinitesimal surface element

*dA*:

*dm*/

*dA*=

*ρdz*= −

*dp*/

*g*across each layer with density

*ρ*, pressure

*p*, and the gravitational acceleration

*g*(note that pressure and density vary throughout the layer). In Bokhove (2002b) and Ripa (1993) the variational and Hamiltonian formulation of the isentropic

*N*-layer equations are derived by simplifying the Eulerian variational principle of the compressible Euler equations.

*α*= 1,2 and in which

*α*, and

*f*is the Coriolis frequency, and

*M*is the Montgomery potential. To close the system, one needs to specify the Montgomery potentials in each layer. As seen in section 3 of Bokhove and Oliver (2009), for a two-layer model these potentials can be defined as

_{α}*κ*=

*R*/

*c*is the ratio between the specific gas constant for dry air (

_{p}*R*= 287 J kg

^{−1}K

^{−1}) and its specific heat capacity at constant pressure (

*c*= 1004 J kg

_{p}^{−1}K

^{−1}).

*M*/∂

*z*= 0 implies that, in general, the Montgomery potential

*M*=

*c*+

_{p}θη^{κ}*gz*is independent of

*z*within each layer. Therefore, one can evaluate

*M*in the bottom layer (where

*θ*=

*θ*

_{2}) at both

*z*=

*z*

_{2}and

*z*=

*z*

_{1}, and

*M*in the upper layer (where

*θ*=

*θ*

_{1}) at both

*z*=

*z*

_{1}and

*z*=

*z*

_{0}, to find

*η*

_{0}is treated as a constant throughout the paper.

*σ*

_{1}and

*σ*

_{2}as follows:

*M*

_{1}in (3c), leading to

*gZ*

_{0}from both sides one finds

If the top surface is fixed, i.e., *z*_{0} = *Z*_{0}, then *M*_{1} = *g*(*z*_{0} − *Z*_{0}) = 0, and a closed 1½-layer model emerges as follows (a sketch of the model is given in Figs. 1b,d). For the 1½-layer model, the momentum equations in the lower stratospheric layer remain as in Eqs. (3a) and (3b) for *α* = 2. The model is indeed closed, because *M*_{1} = 0 defines *η*_{1} = *p*_{1}/*p _{r}* in terms of

*η*

_{2}=

*p*

_{2}/

*p*. This fact allows

_{r}*σ*

_{2}to be expressed in terms of

*η*

_{2}as follows:

*σ*

_{2}(

*η*

_{2}) =

*p*[

_{r}*η*

_{2}−

*η*

_{1}(

*η*

_{2})]/

*g*. We note that such a 1½-layer model has the advantage over a one-layer model that the pressure

*p*

_{1}is active and not constrained to be constant, as is

*p*

_{0}. Consequently, the values of the surface pressure

*p*

_{2}are more realistic. At first sight, the 1½-layer model, however, seems inconsistent, since the constraint

*M*

_{1}= 0 is not preserved in time by the original two continuity equations. Nevertheless—as we will show later in this paper—the closed 1½-layer model [(3a) and (3b)] with

*α*= 2 and Montgomery potential

*M*

_{2}results after taking

*M*

_{1}= 0 and

**v**

_{1}= 0 in the momentum equation of the stratospheric layer. Perhaps not surprisingly, the original potential energy of the two-layer model subject to the constraint

*M*

_{1}=

*g*(

*z*

_{0}−

*Z*

_{0}) = 0 does give the desired potential energy of the 1½-layer model.

## 3. Scaling of a two-layer model and asymptotic analysis

### a. Nondimensionalization and scaling of the two-layer model

*L*is the horizontal length scale;

*U*and

_{α}*H*are the layer velocity and depth scale, respectively; Σ

_{α}_{1}is a constant; Fr

*indicates the layer Froude number*

_{α}*c*

_{p}θ_{1}/

*gH*

_{1}is assumed to be

*ε*is the layer velocity ratio

*ε*=

*U*

_{1}/

*U*

_{2};

*δ*=

_{a}*H*

_{2}/

*H*

_{1}defines the layer thickness ratio. Both

*ε*and

*δ*are assumed to be small and the assumption made in (9) implies a scaling on

_{a}*δ*via the Froude number. We will discuss in section 4 to what extent the chosen scaling is supported by observations.

_{a}*η*

_{0}is small compared to (nondimensional)

*σ*

_{1}. Again, the validity of the latter on observational evidence is discussed in section 4; nevertheless, even in absence of such hypothesis, the rest of the derivation would only differ for the presence of an additional constant.

*being the layer Rossby number*

_{α}*M*

_{1}. Therefore, we define the (constant) mean potentials

*ε*

^{2}inside

*δ*in (9) has been used.

_{a}As will appear clearer in both the asymptotic analysis and the Hamiltonian derivation, the mean potentials *ε*.

### b. Asymptotic analysis

*ε*—that is, at

*σ*

_{2}= 0):

We introduce a fast time scale *τ* = *t*/*ε* and evaluate (15) at leading order; that is, we truncate the system (15) and (16) at the fast time scale by taking the limit *ε* → 0 (after multiplication by *ε*).

*D*

_{1}oscillate rapidly, while the slow variables

*ω*

_{1},

*σ*

_{2}, and

**v**

_{2}vary on the slow time scale. The introduction of fast and slow variables is based on the distinction between high-frequency and low-frequency waves in linearized wave equations (Van Kampen 1985). Later in section 5, we will consider the reduced, Hamiltonian dynamics on the “slow” manifold defined by the constraints (17).

## 4. Observations supporting the scaling

The validity of the scaling used in section 3 determines whether the 1½-layer model to be derived in this paper is suitable to represent the real atmosphere. In this sense, here we show two possible applications: one in the stratosphere and one in the troposphere, with the latter being more relevant for the purpose of Part I of this study, as convection and precipitation are confined therein. Table 1 summarizes the values derived from real atmospheric measurements (i.e., from radiosonde data) of the main relevant physical quantities together with the associated nondimensional parameters *δ _{a}*, and

*ε*. A sketch of the model configuration for both cases is shown in Fig. 1.

Summary of the values of various physical quantities obtained from the radiosonde data displayed in Figs. 2 and 3 and resulting values of nondimensional scaling parameters *δ _{a}* and

*ε*. The values of Fr

_{1}and Fr

_{2}—scaling hypotheses made earlier in section 3—are also reported. The values of

*p*

_{1}and

*θ*

_{1}in the bottom rows are computed via (5) using the observed quantities

### a. Two-layer stratosphere

The scaling is compatible with a 1½-layer approximation of the stratosphere. In this regard, our estimates are based on zonally averaged climatological seasonal radiosonde data displayed in Birner (2006), where potential temperature and horizontal wind speed are displayed as function of height and latitude. On request, Dr. Birner extracted vertical profiles of potential temperature and pressure at about 57.25°N (i.e., at midlatitudes) versus height in the summer season, shown in Fig. 2. From the data, it seems reasonable to take *Z*_{2} = 10.6 km for the tropopause height (the lower bound) and, say, *Z*_{1} = 16.6 km and *Z*_{0} = 34.6 km. Hence, *η*_{0} M *σ*_{1}. In our scaling, *ε*, *δ _{a}*,

*θ*

_{1}, and

*p*

_{1}follow, for example, after choosing

*g*= 9.81 ms

^{−1},

*c*= 1004.6 J kg

_{p}^{−1}K

^{−1},

*R*= 287.04 J kg

^{−1}K

^{−1}, and

*p*= 1000 mb such that

_{r}*κ*= 2/7. We obtain,

*ε*≈ 0.14,

*δ*≈ 0.33,

_{a}*θ*

_{1}= 629 K [solving (5a) for

*θ*

_{1}], and

*p*

_{1}= 97 mb [solving (5b) for

*η*

_{1}and hence

*p*

_{1}]. All data are reported in Table 1. This pressure value compares well with the observed one, while the calculated and observed potential temperature differ somewhat because the observed buoyancy frequency (and temperature) is roughly constant and not the entropy as in our layer model. The values of the Froude numbers Fr

_{1}and Fr

_{2}deduced from the scaling parameters

*ε*and

*δ*are within a factor 5 from those deduced from the observations

_{a}*c*

_{p}θ_{1}/(

*gH*

_{1}) = 3.58. Despite these slight differences, the data provide an observational basis for the chosen scaling.

Profiles, zonally averaged, (solid lines) of (a) observed potential temperature *θ*(*z*) and (b) pressure *p*(*z*) vs height *z* at circa 57.25°N, and extrapolated profiles (dash–dotted lines) from *Z*_{1} = 16.63 km to *Z*_{0} = 34.63 km based on the approximately constant scale heights of the observed *θ* and *p*, respectively, in the stratosphere. The tropopause lies at approximately *Z*_{2} = 10.63 km. Data courtesy Dr. Thomas Birner (cf. Birner 2006). The relevant physical parameters associated with these vertical profiles are reported in Table 1.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

Profiles, zonally averaged, (solid lines) of (a) observed potential temperature *θ*(*z*) and (b) pressure *p*(*z*) vs height *z* at circa 57.25°N, and extrapolated profiles (dash–dotted lines) from *Z*_{1} = 16.63 km to *Z*_{0} = 34.63 km based on the approximately constant scale heights of the observed *θ* and *p*, respectively, in the stratosphere. The tropopause lies at approximately *Z*_{2} = 10.63 km. Data courtesy Dr. Thomas Birner (cf. Birner 2006). The relevant physical parameters associated with these vertical profiles are reported in Table 1.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

Profiles, zonally averaged, (solid lines) of (a) observed potential temperature *θ*(*z*) and (b) pressure *p*(*z*) vs height *z* at circa 57.25°N, and extrapolated profiles (dash–dotted lines) from *Z*_{1} = 16.63 km to *Z*_{0} = 34.63 km based on the approximately constant scale heights of the observed *θ* and *p*, respectively, in the stratosphere. The tropopause lies at approximately *Z*_{2} = 10.63 km. Data courtesy Dr. Thomas Birner (cf. Birner 2006). The relevant physical parameters associated with these vertical profiles are reported in Table 1.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

### b. Two-layer troposphere in presence of a low-level jet

The scaling presented in section 3 is also compatible with a 1½-layer approximation in the troposphere. LLJs are recurrent meteorological features located at various locations in the world (Rife et al. 2010) and they happen to be particularly common over the Great Plains in the southern United States (Ladwig 1980; Djurić and Damiani 1980).

Figure 3 shows vertical profiles obtained from radiosonde data of both potential temperature and wind speed during an LLJ event on 10–11 December 1977 in Brownsville, Texas (United States). We use this as a case study to provide a justification for the scaling chosen in section 3. We approximate the troposphere as a two-layer fluid, exploiting the discontinuities in the potential temperature profile of Fig. 3b as a reference. Mean potential temperature values of *η*_{0} > *σ*_{1}. All in all, it is possible to see from Table 1 how *ε* and *δ _{a}* lie below one during the LLJ event; moreover, the rigid-lid condition leading to the 1½-layer configuration appears to be justified, as the variation in height of

*Z*

_{0}=

*H*

_{1}+

*H*

_{2}= 6, 6.1, and 6.25 km is smaller than the change in depth of the bottom layer

*H*

_{2}= 2.02, 2.08, and 1.65 km. Furthermore, the values of

*p*

_{1}and

*θ*

_{1}computed via Eq. (5) using the observed values

_{1}and Fr

_{2}deduced from the scaling parameters

*ε*and

*δ*are on average (cf. rightmost column in Table 1) within a factor of 8 from those deduced from the observations

_{a}Vertical profile of potential temperature (solid line) and wind speed (dashed line) taken from radiosonde data at (a) 0000 UTC 10 Dec 1977, (b) 1200 UTC 10 Dec 1977, and (c) 0000 UTC 11 Dec 1977 in Brownsville. The horizontal dotted lines indicate the depth of the two layers deduced from potential temperature data. The relevant physical parameters associated with each vertical profile are reported in Table 1. Source: http://weather.uwyo.edu/upperair/sounding.html.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

Vertical profile of potential temperature (solid line) and wind speed (dashed line) taken from radiosonde data at (a) 0000 UTC 10 Dec 1977, (b) 1200 UTC 10 Dec 1977, and (c) 0000 UTC 11 Dec 1977 in Brownsville. The horizontal dotted lines indicate the depth of the two layers deduced from potential temperature data. The relevant physical parameters associated with each vertical profile are reported in Table 1. Source: http://weather.uwyo.edu/upperair/sounding.html.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

Vertical profile of potential temperature (solid line) and wind speed (dashed line) taken from radiosonde data at (a) 0000 UTC 10 Dec 1977, (b) 1200 UTC 10 Dec 1977, and (c) 0000 UTC 11 Dec 1977 in Brownsville. The horizontal dotted lines indicate the depth of the two layers deduced from potential temperature data. The relevant physical parameters associated with each vertical profile are reported in Table 1. Source: http://weather.uwyo.edu/upperair/sounding.html.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0023.1

## 5. Hamiltonian derivation

### a. Constrained Hamiltonian formulation of the 1½-layer equations

*α*= 1, 2):

*q*in each layer

_{α}*α*:

*σ*

_{1}, arises without problem because any multiple of the mass

*M*

_{1}=

*g*(

*z*

_{0}−

*Z*

_{0}), which is a useful simplification as we have seen.

**v**

*= (*

_{α}*u*,

_{α}*υ*). The second part of (24) follows from (22) after some calculation using the definitions of the Montgomery potentials in (3c) and (3d) and the pseudodensity (6) in (3).

_{α}In the verification of the properties above (as well as in some others later in the paper), certain integral terms vanish in presence of certain boundary conditions such as periodic boundaries; quiescence and constancy at infinity where *σ _{α}* is constant and

**v**

*= 0; slip flow along walls, such that*

_{α}*L*

^{2}, with potential vorticities:

*σ*

_{2}, are added to obtain a Hamiltonian of

*ε*→ 0. These extra terms arise because mass is globally conserved in each layer and can be introduced formally by adding constants and mass Casimirs

*λ*

_{1}and

*λ*

_{2}(more details are given in appendix A). These above Casimirs are conserved since

*J*(

*a*,

*b*) := (∂

*)(∂*

_{x}a*) − (∂*

_{y}b*)(∂*

_{x}b*) being the Jacobian operator. Note that, from (27), it follows that*

_{y}a*q*is

_{α}*D*

_{1},

*ω*

_{1},

**v**

_{2},

*σ*

_{2}):

*ε*the variational derivative of the Hamiltonian, (28), is (see appendix C):

*χ*and (transport) streamfunction Ψ. Therefore, using (33) one finds

*ε*one obtains

*ε*we take

*ε*.

*ω*

_{1}(

*x*,

*y*, 0) = 0. Together with

*D*

_{1}= 0, this explains why it is asymptotically allowed to take

**v**

_{1}= 0 at leading order, as we discussed at the end of section 2.

**v**

_{1}= 0]:

*ε*and using the (higher-order) constraint

**v**

_{1}= 0 [by initializing

*ω*

_{1}(

*x*,

*y*, 0) = 0] and

*ε*. When we truncate this higher-order Hamiltonian one finds again

**v**

_{1}= 0 and

*z*

_{0}=

*Z*

_{0}to get the rigid-lid approximation

*ω*= 0,

**v**

_{2},

*σ*

_{2}}, since the dynamics of the fast variables {

*D*

_{1},

*D*

_{1},

*φ*

_{1}} associated with the gravity waves in the top layer is absent at leading order. Restricting or truncating the transformed bracket (31) to the constrained manifold and keeping all leading-order terms in

*ε*, the following (dimensional and dimensionless) constrained dynamics emerges:

**v**

_{2}and

*σ*

_{2}only.

### b. The Jacobi identity

The bracket (40) satisfies the Jacobi identity since it coincides with the bottom-layer terms in the original bracket, (26), which consists of two uncoupled parts, one for each layer, to which the Jacobi identity can be applied separately. In this sense, the preservation of the Jacobi identity for the leading-order reduced bracket, (40) is straightforward to prove in the asymptotic analysis presented in this paper. However, proving the Jacobi identity for the leading-order reduced bracket resulting from a singular perturbation approach in the general case is more complicated and we refer to Bokhove (1996, 2002a) for a more extensive discussion of this topic.

### c. Dimensional dynamics

*σ*

_{2}= (

*p*

_{2}−

*p*

_{1})/

*g*,

*σ*

_{1}=

*p*

_{1}/

*g*, and the constraint

*M*

_{1}= 0 (i.e.,

*z*

_{0}=

*Z*

_{0}) relating

*η*

_{1}=

*p*

_{1}/

*p*to

_{r}*η*

_{2}=

*p*

_{2}/

*p*, that is,

_{r}*ε*, we use the original Hamiltonian reduced to the constraint, or “rigid-lid” manifold,

*M*

_{1}= 0 (and

**v**

_{1}= 0). Hence, we include higher-order terms in

*ε*in the Hamiltonian. This does not hamper the leading-order accuracy since the constrained bracket (40) is leading order. The functional derivative of the potential and internal energy in (41) subject to constraint (42) is

*gσ*

_{2}=

*p*

_{2}−

*p*

_{1}and with

*α*= 2 thus stay the same with Montgomery potential (3c), in which

*σ*

_{1}is defined in terms of

*σ*

_{2}and

*z*

_{2}by

*M*

_{1}= 0 via (3d).

Recapitulating, we note that we have been able to construct the Hamiltonian formulation of an isentropic 1½-layer model. Importantly, we conclude a posteriori that it is consistent to set **v**_{1} = 0, since in the upper layer we found *ω*_{1} = 0 by initializing *ω*_{1}(*x*, *y*, 0) = 0 and *D*_{1} = 0 in the small *ε* limit.

## 6. Conclusions

In this paper, Part II, we have provided a full mathematical derivation of the isentropic 1½-layer shallow-water model utilized in Part I. Starting from an isentropic two-layer model, we show how a rigid-lid constraint alone (leading to the condition on the Montgomery potential in the top layer *M*_{1} = 0) does not suffice to derive an entirely consistent 1½-layer model, resulting instead in an apparent inconsistent configuration due to the nonpreservation in time of the *M*_{1} = 0 constraint.

To resolve this apparent inconsistency, we have shown how the 1½-layer model emerges from the two-layer one, once the latter is properly scaled to allow for the asymptotic analysis. In the limit *ε* = *U*_{1}/*U*_{2} → 0, two constraints emerge, i.e., *M*_{1} = 0 and **v**_{1} = 0, and the system reduces to a single set of equations for the (slow) variables in the bottom layer.

We have further demonstrated that the scaling used in the asymptotic analysis can be justified on the basis of real observations; these arise in both the modeling of a two-layer stratosphere and that of a two-layer troposphere in the presence of a low-level jet. The latter is the most useful in view of using the idealized model described in Part I for satellite data assimilation research.

Finally, a Hamiltonian derivation of the model has been undertaken in section 5, where a slaved Hamiltonian approach has been used—generalized here for the infinite-dimensional case—thus removing an apparent inconsistency in the model derivation as well as underpinning the conservative nature of the system.

## Acknowledgments.

This work stems in part from the work done by Luca Cantarello under a NERC SPHERES DTP scholarship (NE/L002574/1, Reference 1925512), cofunded by the Met Office via a CASE partnership. We thank the Deputy Chief Meteorologist Nicholas Silkstone (Met Office) for his useful suggestion regarding the low-level jet conditions that has guided our choice on a suitable scaling for our model in the troposphere. We also thank Gordon Inverarity (Met Office), Prof. Rupert Klein (Freie Universität Berlin), and another anonymous reviewer for their constructive comments. Finally, our thanks go to Prof. Thomas Birner for the discussion and use of his data (2006/07). We do not have conflicts of interest to disclose.

## Data availability statement.

The stratospheric observational data used in section 4 can be found in the referenced paper (Birner 2006), while the tropospheric ones are obtained from the University of Wyoming’s Atmospheric Soundings web page (see http://weather.uwyo.edu/upperair/sounding.html).

## APPENDIX A

### Scaled Hamiltonian of a Two-Layer Shallow-Water Model

*ε*→ 0, that is,

*σ*

_{2}= 0) and by retaining only the terms at leading order in

*ε*, with

*ε*→ 0.

## APPENDIX B

### Change of Variables in the Functional Derivatives

*σ*

_{2},

**v**

_{1},

**v**

_{2}) in terms of those derived in the asymptotic analysis (

*D*

_{1},

*ω*

_{1},

*φ*

_{1},

*σ*

_{2},

**v**

_{2}). We start from the definition of the differential

*D*

_{1}and

*ω*

_{1}are functions of

**v**

_{1}, we can write

*δD*

_{1}and

*δω*

_{1}have been computed as

The functional derivatives with respect to the other variables can be obtained accordingly.

## APPENDIX C

### Scaled Variational Hamiltonian in the Limit ε → 0

*χ*and a streamfunction Ψ, i.e.,

*δD*

_{1}and

*δω*

_{1}in (B4) have been used.

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